Jeremy Borden | Searcharxiv

arXiv2026

Searching for the Proton's Missing Spin: Small-$x$ Helicity Evolution Equations and Their Analytic Solutions

The proton spin puzzle denotes the challenge of describing the proton's spin in terms of the angular momenta of the quarks and gluons which comprise it. These quarks and gluons carry a fraction $x$ of the proton's momentum. Contributions from small-$x$ quarks and gluons, which only possess a little of the proton's momentum, are difficult to measure, since this requires very high energy experiments. Furthermore, early theoretical work in the 1990s predicted substantial contributions to the proton spin from these small-$x$ particles. We need theoretical control over this corner of phase space in order to resolve the spin puzzle. In this dissertation, we build upon an existing framework for studying spin at small-$x$. Previously, several sets of small-$x$ evolution equations were derived in this formalism -- one in the large-$N_c$ limit and one in the large-$N_c\&N_f$ limit. Here $N_c$ and $N_f$ are the numbers of quark colors and flavors. These equations were numerically solved but no analytic solutions had been found. In this dissertation we detail the construction of such analytic solutions, first in the large-$N_c$ limit and then in the large-$N_c\&N_f$ limit, after deriving an important correction to the existing large-$N_c\&N_f$ equations due to the contributions of quark-to-gluon transition operators. From the solutions constructed here, we can predict the behavior of the quark and gluon helicity distributions at asymptotically small-$x$ (and large-$N_c$ or large-$N_c\&N_f$), both as a general power law and further as explicit analytic expressions in the asymptotic limit. Our solutions also allow us to predict all four polarized DGLAP anomalous dimensions in the same limits, yielding expressions exact to all orders in the strong coupling. The expansions of our predictions agree completely with the full extent of existing finite-order calculations, to three loops.

hep-ph↗

arXiv2025

Analytic Solution for the Helicity Evolution Equations at Small $x$ and Large $N_c\&N_f$

We construct an exact analytic solution of the revised small-$x$ helicity evolution equations, where the contributions of the quark-to-gluon and gluon-to-quark transition operators were newly included. These evolution equations are written in the large-$N_c\&N_f$ limit and are double-logarithmic, resumming powers of $α_s\ln^2(1/x)$. Here $N_c$ and $N_f$ are the numbers of quark colors and flavors, while $α_s$ is the strong coupling constant and $x$ is the Bjorken-$x$ variable. Using our solution, we obtain analytic expressions for the flavor singlet quark and gluon helicity parton distribution functions (PDFs) and for the $g_1$ structure function as double-inverse Laplace transforms. We also extract analytic expressions for the four DGLAP polarized anomalous dimensions $Δγ_{qq}, Δγ_{qG}, Δγ_{Gq}$, and $Δγ_{GG}$: these expressions resum powers of $α_s/ω^2$ to all orders at large-$N_c\&N_f$ (with $ω$ the Mellin moment variable). We extract the leading small-$x$ growth of the helicity distributions, \begin{align} ΔΣ(x,Q^2) \sim ΔG(x,Q^2)\sim g_1(x,Q^2) \sim \left(\frac{1}{x}\right)^{α_h}, \end{align} where the intercept $α_h$ satisfies an algebraic equation. We determine $α_h$ numerically for various values of $N_c$ and $N_f$. We further obtain the explicit asymptotic expressions for the helicity distributions, which yield numerical values for the ratio of the gluon helicity PDF to the flavor singlet quark helicity PDF in the small-$x$ asymptotic limit (for different $N_f/N_c$). We find that all our predictions for polarized DGLAP anomalous dimensions are fully consistent with the existing finite-order calculations. Similar to the large-$N_c$ case, our intercept $α_h$ exhibits a very slight disagreement with the predictions made within the infrared evolution equations framework.

hep-ph↗

arXiv2024

Helicity Evolution at Small $x$: Quark to Gluon and Gluon to Quark Transition Operators

We include the quark to gluon and gluon to quark shock-wave transition operators into the small Bjorken-$x$ evolution equations for helicity in the flavor-singlet channel derived earlier. While such transitions do not affect the large-$N_c$ version of the evolution equations for helicity, the large-$N_c \& N_f$ equations are affected. ($N_c$ and $N_f$ are the numbers of quark colors and flavors, respectively.) We derive the corresponding corrected large-$N_c \& N_f$ equations for the polarized dipole amplitudes contributing to the flavor-singlet quark and gluon helicity distributions in the double-logarithmic approximation (DLA), resumming powers of $α_s \, \ln^2 (1/x)$ with $α_s$ the strong coupling constant. We solve these equations iteratively and extract the polarized splitting functions up to four loops. We show that our splitting functions agree with the fixed-order perturbative calculations up to and including the existing three-loops results. Similar to the large-$N_c$ helicity evolution in the shock-wave approach, our large-$N_c \& N_f$ small-$x$ splitting functions agree with those obtained in the infrared evolution equations framework up to three loops, but appear to slightly disagree at four loops.

hep-ph↗

arXiv2023

Analytic Solution for the Revised Helicity Evolution at Small $x$ and Large $N_c$: New Resummed Gluon-Gluon Polarized Anomalous Dimension and Intercept

We construct an exact analytic solution of the revised small-$x$ helicity evolution equations derived recently. The equations we solve are obtained in the large-$N_c$ limit (with $N_c$ the number of quark colors) and are double-logarithmic (summing powers of $α_s \ln^2(1/x)$ with $α_s$ the strong coupling constant and $x$ the Bjorken $x$ variable). Our solution provides small-$x$, large-$N_c$ expressions for the flavor-singlet quark and gluon helicity parton distribution functions (PDFs) and for the $g_1$ structure function, with their leading small-$x$ asymptotics given by \begin{align} ΔΣ(x, Q^2) \sim ΔG (x, Q^2) \sim g_1 (x, Q^2) \sim \left( \frac{1}{x} \right)^{α_h} , \notag \end{align} where the exact analytic expression we obtain for the intercept $α_h$ can be approximated by $α_h = 3.66074 \, \sqrt{\frac{α_s \, N_c}{2 π}}$. Our solution also yields an all-order (in $α_s$) resummed small-$x$ anomalous dimension $Δγ_{GG} (ω)$ which agrees with all the existing fixed-order calculations (to three loops). Notably, our anomalous dimension is different from that obtained in the infrared evolution equation framework developed earlier by Bartels, Ermolaev, and Ryskin (BER), with the disagreement starting at four loops. Despite the previously reported agreement at two decimal points based on the numerical solution of the same equations, the intercept of our large-$N_c$ helicity evolution and that of BER disagree beyond that precision, with the BER intercept at large $N_c$ given by a different analytic expression from ours with the numerical value of $α_h^{BER} = 3.66394 \, \sqrt{\frac{α_s \, N_c}{2 π}}$. We speculate on the origin of this disagreement.

hep-ph↗

arXiv2023

Analytic Solution for the Revised Helicity Evolution at Small $x$ and Large $N_c$: New Resummed Gluon-Gluon Polarized Anomalous Dimension and Intercept

We construct an exact analytic solution of the revised small-$x$ helicity evolution equations derived recently. The equations we solve are obtained in the large-$N_c$ limit (with $N_c$ the number of quark colors) and are double-logarithmic (summing powers of $α_s \ln^2(1/x)$ with $α_s$ the strong coupling constant and $x$ the Bjorken $x$ variable). Our solution provides small-$x$, large-$N_c$ expressions for the flavor-singlet quark and gluon helicity parton distribution functions (PDFs) and for the $g_1$ structure function, with their leading small-$x$ asymptotics given by \begin{align} ΔΣ(x, Q^2) \sim ΔG (x, Q^2) \sim g_1 (x, Q^2) \sim \left( \frac{1}{x} \right)^{α_h} , \notag \end{align} where the exact analytic expression we obtain for the intercept $α_h$ can be approximated by $α_h = 3.66074 \, \sqrt{\frac{α_s \, N_c}{2 π}}$. Our solution also yields an all-order (in $α_s$) resummed small-$x$ anomalous dimension $Δγ_{GG} (ω)$ which agrees with all the existing fixed-order calculations (to three loops). Notably, our anomalous dimension is different from that obtained in the infrared evolution equation framework developed earlier by Bartels, Ermolaev, and Ryskin (BER), with the disagreement starting at four loops. Despite the previously reported agreement at two decimal points based on the numerical solution of the same equations, the intercept of our large-$N_c$ helicity evolution and that of BER disagree beyond that precision, with the BER intercept at large $N_c$ given by a different analytic expression from ours with the numerical value of $α_h^{BER} = 3.66394 \, \sqrt{\frac{α_s \, N_c}{2 π}}$. We speculate on the origin of this disagreement.

hep-ph↗